1. Harmonics, Timbre & The Frequency Domain

2. Real Instruments Real instruments do not normally produce pure tones
Instead the sound produced by hitting a single note has:
a fundamental frequency
some extra frequencies

3. Harmonics Frequencies can be harmonically related
These are called harmonics
They are related in whole number multiples

4. Harmonics
If extra frequencies are harmonics the sound produced will be perceived as a single pitch at the fundamental frequency
One fundamental frequency and several harmonics

5. Harmonics The harmonics will be multiples of the fundamental frequency
The fundamental is the largest common divisor of the harmonics

7. Harmonics The composite wave has the same frequency as the fundamental
The fundamental is frequency at which the entire waveform vibrates
The brain perceives the composite waveform as a sound that has the same pitch as the fundamental

8. Fundamental Tracking
The ability of human brain to track the fundamental frequency of a sound
Occurs even when the fundamental waveform is not present
This is because the wave will repeat at the fundamental frequency and the brain detects this

9. f0 = 1/2f1 = 1/3f2 f1 f0 pressure f2 time t2 t1 t0 2.5 2 1.5 1 0.5
-2.5 -2
-1.5 -1
-0.5
0.5 1
1.5 2
2.5
pressure t1 t2 t0 f0 f2 f1
time

10. f1 = 2 and f2 are integer multiples of f0
t0 = 2t1 = 3t2
substitute in t = 1/f to get:
f0 = 1/2f1 = 1/3f2
So: f1 = 2f0 and f2 = 3f0

11. For Example
Say: f1 = 440Hz
1/2f1 = 1/3f2 : f1 = 2/3f2 : 3f1 = 2f2 : f2 = 3/2f1
So: f2 = 3/2 * 440 = 660Hz
f0 = 1/2f1 = 220Hz
Which is the highest common divisor of 440 and 660

12. Harmonic Series

13. Harmonic Series

14. Frequency Domain Representation
Used to sound waves being represented in terms of time and amplitude
Known as Time Domain Representation
Frequency Domain Rep shows frequency and amplitude

15. Frequency Domain Representation1
- 1 dB
seconds 1
- 1 dB
frequency
100
200
300
400
500
600
A 440Hz sine wave shown in the time domain (above) and the frequency domain (below).

16. Frequency Domain Plots
Both real instruments and synthesisers normally produce more complex waves
This means additional frequency components or overtones
All of these frequencies (including the fundamental) are called partials

17. Frequency Domain Plots
The frequency domain content of a wave is represented by plotting each partial on the x-axis
The height of each line indicates the strength of each frequency component

18. Trumpet
amplitude
frequency

19. Clarinet
amplitude
frequency

20. Trumpet

21. Timbre
‘Characteristic quality of sounds produced by each particular voice or instrument, depending on the number and character of overtones.’ Oxford English Dictionary

22. ‘A common timbre groups tones played by an instrument at different pitches, loudnesses, and durations. No matter what note it plays, for example, we can always tell when a piano is playing. Human perception separates each instrument’s tones from other instrument tones played with the same pitch, loudness, and duration. No one has much trouble separating a marimba from a violin tone of the same pitch, loudness and duration. Of course a single instrument may also emit many timbres, as in the range of sonorities obtained from saxophones blowed at different intensities.’
Roads (1996, p 544)

23. Harmonics Again
Remember: a harmonic is a sound that is an integer multiple of the fundamental frequency
So: while the fundamental carries out one cycle (or period), a harmonic of this will carry out an exact number of whole cycles

25. Harmonics Again Still
So a fundamental plus several harmonics produces a composite waveform that is?
Periodic: it repeats itself exactly
If the added waveforms were not harmonics the waveform would not be periodic

26. Timbre
‘Characteristic quality of sounds produced by each particular voice or instrument, depending on the number and character of overtones.’ Oxford English Dictionary

27. Harmonics and Musical Instruments
Most instruments produce overtones
Generally the overtones are nearly harmonic but not quite
Because they are not exact harmonics the sound wave produced is not periodic (but quasi-periodic)

28. Harmonics and Musical Instruments
The trumpet has strong harmonics
(brash, full and brassy)
The clarinet has weaker ones
(pure, smooth flute like)
In-harmonic frequencies are non-periodic
(metallic and percussive)

29. Trumpet
amplitude
frequency

30. Clarinet
amplitude
frequency

32. Amplitude Variations
Timbre is also strongly affected by variations in amplitude over time
This is why amplitude envelopes are so commonly used in electronic sound synthesis

33. Trumpet

34. Formants A formant is a peak of energy in an absolute frequency region
Responsible for the timbre of the human voice and many real instruments
Formant peaks stay the same when pitch of sound is changed

35. Wave Shapes
Wave shapes like triangle waves, and sawtooth waves don’t have formants
This is because they have a shape that always remains the same
They have identical relationships among their frequency components no matter what pitch they are

36. Triangle Wave
Amplitude
frequency 1
31
71
51
111
91
131

37. Triangle Wave
Sound produced will be mostly between 1st and 7th harmonics
So one of 70Hz will have it’s most prominent frequencies Hz
And one of 500Hz will have it’s most prominent frequencies Hz

38. Sound Sources with Formants
Like the human voice (e.g. saying ah) will change shape depending on their frequency

39. Humans Saying “Ahh”
On average the following frequencies are emphasised when a man says “ahh”: 730Hz, 1090Hz, and 2440Hz
No matter what the pitch the man says ah at
Female “ahh”s are on average: 850Hz, 1220Hz and 2810Hz

40. Formants Are Caused by Physical Characteristics
The formants of a particular instrument or voice are determined by resonance chambers
Human voice formants vary according to nasal, oral and pharyngeal cavities

41. Formants Are Caused by Physical Characteristics
A guitar has formants based on shape character and dimensions of resonance chamber
Formants are what make a human voice recognisable or give an instrument a particular sound